Example. For \(\mathfrak {sl}_2\), the two Clifford actions have different constituents [fcap-003F]
✍️source

Take the basis \(e,f,h\) of \(\mathfrak {sl}_2(\mathbb C)\) with \[[h,e]=2e,\qquad [h,f]=-2f,\qquad [e,f]=h.\] The Killing form satisfies \(\kappa (e,f)=4\) and \(\kappa (h,h)=8\). Hence the Clifford relations are \[e^2=f^2=0,\qquad ef+fe=8,\qquad h^2=8,\qquad he=-eh,\quad hf=-fh.\] Substitution in the lift of Proposition [fcap-003E] gives \[E=\delta (e)=\frac {he}{8},\qquad F=\delta (f)=\frac {fh}{8},\qquad H=\delta (h)=\frac {ef-fe}{8}=\frac {ef}{4}-1.\] Direct Clifford multiplication yields \[E^2=F^2=0,\qquad H^2=1,\qquad EF=\frac {1+H}{2},\qquad FE=\frac {1-H}{2}.\] Thus \(E,F,EF,FE\) are the four matrix units of the lifted \(\mathfrak {sl}_2\) action.

For left multiplication, the \(H=+1\) and \(H=-1\) subspaces are interchanged by \(F\) and \(E\); the two maps are inverse on the corresponding projected spaces because \(EF\) and \(FE\) are their identity projectors. They have equal dimension, and \(\dim \mathcal {C}\kern -2pt\ell (Q_\kappa )=8\), so each has dimension four. Choosing a basis of the positive space and its \(F\)-images displays \[(\mathcal {C}\kern -2pt\ell (Q_\kappa ),L)\simeq 4V_1,\] four copies of the standard two-dimensional module.

For the commutator action \(D\), antisymmetrization identifies the same vector space with \(\bigwedge \mathfrak {sl}_2\). The exterior powers are \[\bigwedge ^0\mathfrak g\simeq V_0,\qquad \bigwedge ^1\mathfrak g\simeq V_2,\qquad \bigwedge ^2\mathfrak g\simeq V_2,\qquad \bigwedge ^3\mathfrak g\simeq V_0,\] where the middle identification uses the bracket and the last uses the invariant volume. Therefore \[(\mathcal {C}\kern -2pt\ell (Q_\kappa ),D)\simeq 2V_0\oplus 2V_2.\] The dimensions agree but the characters do not. [kostant1997clifford, Theorem 39, pp. 312--313] gives the first answer through Kostant's primary-type theorem. [panyushev2001exterior, Example 2.5(1), internal preprint p. 8] independently gives the exterior decomposition. The displayed matrix-unit calculation directly verifies which action produces which answer.

Example. For \(\mathfrak {sl}_2\), the two Clifford actions have different constituents [fcap-003F]
✍️source

Reference. The exterior algebra and Spin of an orthogonal g-module [panyushev2001exterior]

Reference. Clifford algebra analogue of the Hopf--Koszul--Samelson theorem [kostant1997clifford]

Bivectors and the orthogonal Lie algebra [fcap-000M]

Convention. Bivector normalizations [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000N]

bivector_def bivector_lie_ι
L∃∀N

Definition. Clifford bivector [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000O]

bivectorExterior equivExterior_bivectorExterior
L∃∀N

Lemma. A bivector acts by an infinitesimal rotation [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000P]

bivector_lie_ι
L∃∀N

Proof.

Theorem. Exterior bivectors are the quadratic Clifford Lie algebra [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000Q]

bivectorExteriorEquivQuadraticLieSubalgebra bivectorLieEquiv
L∃∀N

Theorem. Quadratic elements realize the orthogonal Lie algebra [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000R]

soEquivQuadratic soEquivQuadratic_lie_ι
L∃∀N

Example. An elementary skew matrix [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000S]

bivectorEquivSo_apply_ιMulti bivectorEquivSo_apply_ιMulti_mulVec
L∃∀N

Theorem. Reconstructing a skew Clifford lift in polar-dual coordinates [fcap-001Z]

soEquivQuadratic_eq_sum_bivector
L∃∀N

Proof.

Theorem. The Killing-adjoint lift has coefficient one quarter [fcap-0020]

adjointCliffordHom_eq_sum_bivector
L∃∀N

Proof.

Remark. Infinitesimal and global Clifford symmetry [fcap-000T]

Theorem. An orthogonal Lie action lifts to every Clifford module [kostant1997clifford, Section 3.1, pp. 294--295] [fcap-001O]

cliffordInducedRep cliffordInducedRep_apply
L∃∀N

Proposition. Left Clifford multiplication is not the commutator action [fcap-003E]

Example. For \(\mathfrak {sl}_2\), the two Clifford actions have different constituents [fcap-003F]
✍️source

Theorem. The left Killing-Clifford module is primary of type \(V_\rho \) [fcap-003G]

Theorem. The invariant algebra is the internal commutant [fcap-003P]

Remark. Three different commutants for the left Clifford action [fcap-003Q]

Theorem. Primitive invariants and their quadratic form [fcap-003R]

Theorem. Kostant's primitive Clifford factorization [fcap-003S]

Proposition. Normal ordering adds a matrix-trace central character [fcap-0035]

Proposition. The positive CAR product has half-staircase weight [fcap-0036]

Remark. Three different commutants for the left Clifford action [fcap-003Q]

Proposition. Left Clifford multiplication is not the commutator action [fcap-003E]